The Math Notebook
Olympiad problem dissections, AP and SAT analysis, curriculum updates, and exam strategy.
IMO 2024 Problem 1 Dissection: How the Top Students Found the Key Step
The IMO 2024 Problem 1 involved a functional inequality on the real numbers. Most competitors identified the right substitutions; fewer identified the right substitution order. Here is the dissection.
What Changed in Common Core Algebra 2 Assessments in 2024
The 2024 revisions to Common Core Algebra 2 assessment guidance shifted emphasis from procedural fluency to modelling applications, particularly in the statistics and functions units.
SAT Math Digital Format: What Has Changed and What Has Not
The digital SAT math section is shorter, all calculator-permitted, and adaptive. Here is what that means for preparation, score interpretation, and which content topics were emphasised or de-emphasised.
AP Precalculus First Impressions: What the 2024 Exam Revealed
The 2024 AP Precalculus exam was the first under the current course framework. The free-response section surprised many test-takers with its emphasis on polynomial modelling rather than trigonometry.
Why Calculus Students Struggle with Epsilon-Delta Proofs
Epsilon-delta proofs are the formal definition of a limit made rigorous. Most students find them harder than the computations they enable. The difficulty is not mathematical — it is the unusual logical structure.
Mathematics Olympiad Preparation: A Practical Guide for 2025
The mathematics olympiad preparation path is non-linear, and most students underestimate how different competition mathematics is from school mathematics. Here is the structured approach that produces the best results at each level.
AP Statistics: What to Focus on in the Final Six Weeks
Six weeks before the AP Statistics exam is enough time to meaningfully improve your score if you concentrate preparation on the right areas. Inference procedures and interpretive language are where the marks are.
How to Use Desmos Effectively on the Digital SAT
Desmos is not just a graphing calculator — it is an equation solver, system solver, and regression tool. Digital SAT students who know how to use it save 5–8 minutes per section compared to those who treat it as a basic calculator.
The Geometric Intuition Behind Eigenvalues (Finally)
Most introductions to eigenvalues start with the characteristic polynomial and work backwards to the geometric meaning. This one starts with the geometry.
The Real Reason Integration Is Hard (It Is Not What You Think)
Every differentiable function has a unique derivative. Not every continuous function has an antiderivative expressible in elementary functions. This asymmetry is the real source of integration difficulty.
Five Proof Strategies for High School Geometry (When You Cannot See the Next Step)
Geometric proof difficulty is not a measure of your geometry knowledge — it is a measure of whether you have a system. Here are five strategies that recover progress when a proof stalls.
Bayes' Theorem in Practice: Three Problems That Build the Intuition
Bayes' theorem is taught as a formula; it is understood as a way of updating beliefs in the face of evidence. These three problems — of increasing complexity — build the latter from the former.
Why Does Dividing Fractions Work? (The Proof Behind "Invert and Multiply")
"Invert and multiply" is the most memorised rule in fraction arithmetic. Here is why it works — and why knowing why means you can derive it if you forget it.
The Quadratic Formula: Derivation by Completing the Square
The quadratic formula is the most-memorised result in high school algebra. This is the derivation — which means you never need to memorise it again.
What CS Students Actually Need from Discrete Mathematics
Discrete mathematics for CS students is taught as a mathematics course but used as a computer science course. Here are the four areas that show up most directly in programming work.